Well, the only tools I used were a ruler and a protractor to plot points, so the lines are not perfect, but I am pretty sure they should have that curve. Take a look at this old map with the same projection:
File:Novissima totius tarrarum orbis tabula, Auctore Hugonis Allardi.jpg - Wikimedia Commons
There are many like it on Wikimedia Commons and the longitudes and latitudes all look like this.